On fuzzy subnormal and pronormal subgroups of a finite group

Abstract

Abstract Throughout this paper all groups are finite. In 1989, Asaad [2] suggested the following question: knowing some information about a fuzzy subgroup A of a group G , what is the structure of G , and if some information is known about G what is the structure of A ? For example, he proved that if G is a group of prime power order, then G is cyclic iff there exists a fuzzy subgroup A of G such that for x , y in G , (i) if A ( x ) = A ( y ), then 〈 x 〉 = 〈 y 〉 and (ii) if A ( x ) > A ( y ), then 〈 x 〉 ⊂ 〈 y 〉. The aim of this paper is to continue the investigation of this question.